Finite Group Databasebeta

← #13139867

Subgroups — #13139867

Order 1728.

786 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×(C3⋊D9)⋊C4C211yesmaximal, characteristic
2C2×#88536C211yesmaximal, characteristic
3C2×#88445C211yesmaximal, characteristic
4C9×C3⋊#95C211yesmaximal, characteristic, complemented
5C3×C9⋊#95C211yesmaximal, characteristic, complemented
6#11720331C211yesmaximal, characteristic, complemented
7#11720331C211yesmaximal, characteristic, complemented
8#10591371—31nomaximal
9#10588344—31nomaximal
10C23×C3⋊D9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3⋊Dic9C2211yescharacteristic
13C22×C9×Dic3C2211yescharacteristic
14C22×C9×Dic3C2211yescharacteristic
15C22×C3×Dic9C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C2×(C3⋊D9)⋊C4—22no—
18C2×(C3⋊D9)⋊C4—22no—
19C2×(C3⋊D9)⋊C4C414yescomplemented
20#88536—22no—
21#88445—22no—
22C22×#624—31no—
23C2×#3752—31no—
24C2×#3722—31no—
25C22×C4×D9—31no—
26C3×C3⋊#95—31no—
27C3×C3⋊#95D311yescharacteristic
28#63579—31no—
29#63579—31no—
30C2×(C2×D9)⋊C4—31no—
31C2×(C2×D9)⋊C4—31no—
32C2×(C2×D9)⋊C4—61no—
33C9×#95—31no—
34C9⋊#95—31no—
35C9⋊#95—31no—
36C9⋊#95D311yescharacteristic, complemented
37C23×C3×C9C2311yescharacteristic
38C22×C3⋊D9—21no—
39C22×C3⋊D9—21no—
40C22×C3⋊D9—21no—
41C22×C3⋊D9—21no—
42C22×C3⋊D9C2×C412yes—
43C22×C3⋊D9D412yes—
44C22×C3⋊D9D412yes—
45C2×C3×C4×C9—21no—
46C2×C3×C4×C9—21no—
47C2×C3×C4×C9—21no—
48C2×C3⋊Dic9—21no—
49C2×C3⋊Dic9—21no—
50C2×C3⋊Dic9—21no—
51C2×C9×Dic3—21no—
52C2×C9×Dic3—21no—
53C2×C9×Dic3—21no—
54C2×C9×Dic3—21no—
55C2×C9×Dic3—21no—
56C2×C9×Dic3C2×C412yes—
57C2×C3×Dic9—21no—
58C2×C3×Dic9—21no—
59C2×C3×Dic9—21no—
60C2×C3×Dic9—21no—
61C2×C3×Dic9—21no—
62C2×C3×Dic9C2×C412yes—
63(C3⋊D9)⋊C4—28no—
64#4664—91no—
65#4664—91no—
66C23×C3⋊D3—31no—
67C22×C32×C4C2×D311yescharacteristic
68C22×C3⋊Dic3—31no—
69C22×C3×Dic3—31no—
70C22×C3×Dic3—31no—
71C22×C3×Dic3C2×D311yescharacteristic
72C22×C3×Dic3C2×D311yescharacteristic
73C2×#624—34no—
74C2×#624—62no—
75C2×#624—62no—
76C23×D9—31no—
77C23×D9—61no—
78#3752—62no—
79#3722—62no—
80C22×C4×C9—21no—
81C22×C4×C9—31no—
82C22×C4×C9—31no—
83C22×C4×C9C2×D311yescharacteristic
84C22×Dic9—31no—
85C22×Dic9—61no—
86C22×Dic9C2×D311yescharacteristic
87C22×Dic9C2×D311yescharacteristic
88C2×C4×D9—34no—
89C2×C4×D9—62no—
90C2×C4×D9—62no—
91(C2×D9)⋊C4—62no—
92(C2×D9)⋊C4—62no—
93(C2×D9)⋊C4—64no—
94C22×C3×C9C22×C411yescharacteristic
95C22×C3×C9C2×D411yescharacteristic
96C22×C3×C9C2×D411yescharacteristic
97C22×C3×C9#4111yescharacteristic
98C22×C3×C9#4111yescharacteristic
99C22×C3×C9#4111yescharacteristic
100C22×C3×C9#4111yescharacteristic
101C2×C3⋊D9—22no—
102C2×C3⋊D9—22no—
103C2×C3⋊D9—22no—
104C2×C3⋊D9—22no—
105C2×C3⋊D9—22no—
106C2×C3⋊D9—22no—
107C2×C3⋊D9C22⋊C414yes—
108C3×C4×C9—41no—
109C3⋊Dic9—41no—
110C9×Dic3—22no—
111C9×Dic3—41no—
112C3×Dic9—22no—
113C3×Dic9—41no—
114C22×C4×D3—91no—
115C22×C4×D3—91no—
116C2×(C2×D3)⋊C4—91no—
117C2×(C2×D3)⋊C4—91no—
118C2×(C2×D3)⋊C4—91no—
119C2×(C2×D3)⋊C4—91no—
120C2×(C2×D3)⋊C4—181no—
121C3×#95—31no—
122C3×#95—91no—
123C3⋊#95—31no—
124C3⋊#95—91no—
125C3⋊#95—91no—
126C3⋊#95—91no—
127C3⋊#95—91no—
128C3⋊#95D911yescharacteristic, complemented
129C23×C32C22×D311yescharacteristic
130C22×C3⋊D3—32no—
131C22×C3⋊D3—32no—
132C22×C3⋊D3—32no—
133C22×C3⋊D3—61no—
134C22×C3⋊D3—61no—
135C22×C3⋊D3—61no—
136C22×C3⋊D3—61no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C3⋊Dic3—61no—
141C2×C3⋊Dic3—61no—
142C2×C3⋊Dic3—61no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—21no—
145C2×C3×Dic3—21no—
146C2×C3×Dic3—21no—
147C2×C3×Dic3—21no—
148C2×C3×Dic3—32no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—61no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3C4×D312yes—
155#624—68no—
156C23×C9—21no—
157C23×C9C22×D311yescharacteristic
158C22×D9—32no—
159C22×D9—32no—
160C22×D9—32no—
161C22×D9—62no—
162C22×D9—62no—
163C22×D9—62no—
164C22×D9—61no—
165C22×D9—61no—
166C22×D9—61no—
167C22×D9—61no—
168C22×D9—121no—
169C22×D9—121no—
170C22×D9—121no—
171C22×D9—121no—
172C2×C4×C9—22no—
173C2×C4×C9—21no—
174C2×C4×C9—21no—
175C2×C4×C9—21no—
176C2×C4×C9—32no—
177C2×C4×C9—41no—
178C2×C4×C9—41no—
179C2×C4×C9—61no—
180C2×C4×C9—61no—
181C2×C4×C9—61no—
182C2×C4×C9—61no—
183C2×C4×C9—61no—
184C2×Dic9—21no—
185C2×Dic9—21no—
186C2×Dic9—21no—
187C2×Dic9—21no—
188C2×Dic9—21no—
189C2×Dic9—62no—
190C2×Dic9—61no—
191C2×Dic9—61no—
192C2×Dic9—61no—
193C2×Dic9—121no—
194C2×Dic9—121no—
195C2×Dic9C4×D312yes—
196C4×D9—68no—
197#381—271noSylow
198C2×C3×C9C2×C22⋊C411yescharacteristic
199C2×C3×C9#11711yescharacteristic
200C2×C3×C9#11711yescharacteristic
201C2×C3×C9#12311yescharacteristic
202C2×C3×C9#12311yescharacteristic
203C2×C3×C9#12311yescharacteristic
204C2×C3×C9#12311yescharacteristic
205C3⋊D9—24no—
206C23×D3—91no—
207C23×D3—91no—
208C23×D3—181no—
209C22×C3×C4—21no—
210C22×C3×C4—31no—
211C22×C3×C4—31no—
212C22×C3×C4—91no—
213C22×C3×C4—91no—
214C22×C3×C4C2×D911yescharacteristic
215C22×C3×C4D3211yescharacteristic
216C22×Dic3—31no—
217C22×Dic3—31no—
218C22×Dic3—91no—
219C22×Dic3—91no—
220C22×Dic3—181no—
221C22×Dic3C2×D911yescharacteristic
222C22×Dic3C2×D911yescharacteristic
223C2×C4×D3—94no—
224C2×C4×D3—94no—
225C2×C4×D3—182no—
226C2×C4×D3—182no—
227C2×C4×D3—182no—
228C2×C4×D3—182no—
229(C2×D3)⋊C4—182no—
230(C2×D3)⋊C4—182no—
231(C2×D3)⋊C4—182no—
232(C2×D3)⋊C4—182no—
233(C2×D3)⋊C4—184no—
234C22×C32C2×C4×D311yescharacteristic
235C22×C32#23511yescharacteristic
236C22×C32#23511yescharacteristic
237C22×C32D3×D411yescharacteristic
238C22×C32D3×D411yescharacteristic
239C22×C32#23711yescharacteristic
240C22×C32#23911yescharacteristic
241C2×C3⋊D3—34no—
242C2×C3⋊D3—62no—
243C2×C3⋊D3—62no—
244C2×C3⋊D3—62no—
245C2×C3⋊D3—62no—
246C2×C3⋊D3—62no—
247C2×C3⋊D3—62no—
248C32×C4—41no—
249C3⋊Dic3—121no—
250C3×Dic3—22no—
251C3×Dic3—41no—
252C3×Dic3—62no—
253C3×Dic3—121no—
254C22×C9—21no—
255C22×C9—21no—
256C22×C9—21no—
257C22×C9—21no—
258C22×C9—21no—
259C22×C9—21no—
260C22×C9—21no—
261C22×C9C2×C4×D311yescharacteristic
262C22×C9#23511yescharacteristic
263C22×C9#23511yescharacteristic
264C22×C9D3×D411yescharacteristic
265C22×C9D3×D411yescharacteristic
266C22×C9#23711yescharacteristic
267C22×C9#23911yescharacteristic
268C2×D9—34no—
269C2×D9—64no—
270C2×D9—62no—
271C2×D9—62no—
272C2×D9—62no—
273C2×D9—62no—
274C2×D9—62no—
275C2×D9—62no—
276C2×D9—122no—
277C2×D9—122no—
278C2×D9—122no—
279C2×D9—122no—
280C2×D9—122no—
281C2×D9—122no—
282C4×C9—42no—
283C4×C9—41no—
284C4×C9—62no—
285C4×C9—121no—
286Dic9—22no—
287Dic9—41no—
288Dic9—122no—
289Dic9—121no—
290C23×C4—271no—
291C2×C22⋊C4—271no—
292C2×C22⋊C4—271no—
293#95—31no—
294#95—91no—
295#95—271no—
296#95—271no—
297C3×C9#38111yescharacteristic, complemented, Sylow
298C23×C3—21no—
299C23×C3C22×D911yescharacteristic
300C23×C3C2×D3211yescharacteristic
301C22×D3—92no—
302C22×D3—92no—
303C22×D3—92no—
304C22×D3—92no—
305C22×D3—92no—
306C22×D3—92no—
307C22×D3—182no—
308C22×D3—182no—
309C22×D3—182no—
310C22×D3—181no—
311C22×D3—181no—
312C22×D3—181no—
313C22×D3—181no—
314C22×D3—181no—
315C22×D3—181no—
316C22×D3—181no—
317C22×D3—181no—
318C22×D3—361no—
319C22×D3—361no—
320C22×D3—361no—
321C22×D3—361no—
322C2×C3×C4—22no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—32no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—61no—
333C2×C3×C4—61no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—92no—
338C2×C3×C4—181no—
339C2×C3×C4—181no—
340C2×C3×C4—181no—
341C2×C3×C4—181no—
342C2×C3×C4—181no—
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—32no—
349C2×Dic3—61no—
350C2×Dic3—61no—
351C2×Dic3—61no—
352C2×Dic3—61no—
353C2×Dic3—61no—
354C2×Dic3—182no—
355C2×Dic3—181no—
356C2×Dic3—181no—
357C2×Dic3—181no—
358C2×Dic3—181no—
359C2×Dic3—181no—
360C2×Dic3—181no—
361C2×Dic3—361no—
362C2×Dic3—361no—
363C2×Dic3C4×D912yes—
364C4×D3—188no—
365C4×D3—188no—
366C2×C32#87211yescharacteristic
367C2×C32D3×C22⋊C411yescharacteristic
368C2×C32#88311yescharacteristic
369C2×C32#88311yescharacteristic
370C2×C32#89211yescharacteristic
371C2×C32#89411yescharacteristic
372C2×C32#89411yescharacteristic
373C3⋊D3—64no—
374C2×C9—21no—
375C2×C9—21no—
376C2×C9—21no—
377C2×C9—21no—
378C2×C9—21no—
379C2×C9—21no—
380C2×C9—21no—
381C2×C9#87211yescharacteristic
382C2×C9D3×C22⋊C411yescharacteristic
383C2×C9#88311yescharacteristic
384C2×C9#88311yescharacteristic
385C2×C9#89211yescharacteristic
386C2×C9#89411yescharacteristic
387C2×C9#89411yescharacteristic
388D9—64no—
389D9—124no—
390C24—271no—
391C22×C4—31no—
392C22×C4—31no—
393C22×C4—91no—
394C22×C4—91no—
395C22×C4—274no—
396C22×C4—271no—
397C22×C4—542no—
398C22×C4—542no—
399C22×C4D3×D911yescharacteristic
400C22⋊C4—542no—
401C22⋊C4—542no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3C2×C4×D911yescharacteristic
410C22×C3#369711yescharacteristic
411C22×C3#369711yescharacteristic
412C22×C3D4×D911yescharacteristic
413C22×C3D4×D911yescharacteristic
414C22×C3#369911yescharacteristic
415C22×C3#370111yescharacteristic
416C22×C3#379911yescharacteristic
417C22×C3#379911yescharacteristic
418C22×C3C4×D3211yescharacteristic
419C22×C3#380211yescharacteristic
420C22×C3#380411yescharacteristic
421C22×C3#380411yescharacteristic
422C22×C3#381111yescharacteristic
423C2×D3—94no—
424C2×D3—94no—
425C2×D3—184no—
426C2×D3—182no—
427C2×D3—182no—
428C2×D3—182no—
429C2×D3—182no—
430C2×D3—182no—
431C2×D3—182no—
432C2×D3—182no—
433C2×D3—182no—
434C2×D3—182no—
435C2×D3—182no—
436C2×D3—182no—
437C2×D3—182no—
438C2×D3—362no—
439C2×D3—362no—
440C2×D3—362no—
441C2×D3—362no—
442C2×D3—362no—
443C2×D3—362no—
444C3×C4—42no—
445C3×C4—41no—
446C3×C4—41no—
447C3×C4—62no—
448C3×C4—121no—
449C3×C4—182no—
450C3×C4—361no—
451Dic3—22no—
452Dic3—41no—
453Dic3—62no—
454Dic3—121no—
455Dic3—362no—
456Dic3—361no—
457Dic3—361no—
458C32#466411yescharacteristic
459C9—21no—
460C9#466411yescharacteristic, complemented
461C23—272no—
462C23—272no—
463C23—272no—
464C23—541no—
465C23—541no—
466C23—541no—
467C23—541no—
468C23C2×D3×D911yescharacteristic
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—32no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—61no—
476C2×C4—61no—
477C2×C4—61no—
478C2×C4—92no—
479C2×C4—181no—
480C2×C4—181no—
481C2×C4—181no—
482C2×C4—181no—
483C2×C4—181no—
484C2×C4—548no—
485C2×C4—541no—
486C2×C4—541no—
487C2×C4—541no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3#6343411yescharacteristic
496C2×C3C22⋊C4×D911yescharacteristic
497C2×C3#6344511yescharacteristic
498C2×C3#6344511yescharacteristic
499C2×C3#6345411yescharacteristic
500C2×C3#6345611yescharacteristic
501C2×C3#6345611yescharacteristic
502C2×C3#6383611yescharacteristic
503C2×C3#6383611yescharacteristic
504C2×C3#6385411yescharacteristic
505C2×C3#6385411yescharacteristic
506C2×C3#6389711yescharacteristic
507C2×C3#6392111yescharacteristic
508C2×C3#6397511yescharacteristic
509D3—184no—
510D3—184no—
511D3—364no—
512C22—274no—
513C22—542no—
514C22—542no—
515C22—542no—
516C22—542no—
517C22—542no—
518C22—542no—
519C22#8863511yescharacteristic
520C22#8864211yescharacteristic
521C22C4×D3×D911yescharacteristic
522C22#8864611yescharacteristic
523C22#8865811yescharacteristic
524C22#8866211yescharacteristic
525C22#8866811yescharacteristic
526C4—41no—
527C4—62no—
528C4—121no—
529C4—182no—
530C4—361no—
531C4—1081no—
532C3—21no—
533C3#1058834411yescharacteristic, complemented
534C3#1059137111yescharacteristic
535C2—544no—
536C2#1172089811yescharacteristic
537C2#1172089911yescharacteristic
538C2#1172094211yescharacteristic
539C2#1172095811yescharacteristic
540C2#1172099711yescharacteristic
541C2#1172104811yescharacteristic
542C2#1172127511yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC22×C3×C9C22×C4
FrattiniC23×C3C2×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC23×C3⋊D9C22
cosocleC23×C3×C9C23
hypercenterC22×C4D3×D9
hypocenterC3×C9#381
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C9×C3⋊#95C2semidirect
5. C3×C9⋊#95C2semidirect
6. #11720331C2semidirect
7. #11720331C2semidirect
19. C2×(C3⋊D9)⋊C4C4semidirect
36. C9⋊#95D3semidirect
128. C3⋊#95D9semidirect
297. C3×C9#381semidirect
460. C9#4664semidirect
533. C3#10588344semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×(C3⋊D9)⋊C42
C2×#885362
C2×#884452
C9×C3⋊#952
C3×C9⋊#952
#117203312
#105913713
#105883443